In this paper it is investigated as to when a nonempty invariant closed subset A of a -space X containing the set of stationary points (S) can be the fixed point set of an equivariant continuous selfmap on X and such space X is said to possess the S-equivariant complete invariance property (S-ECIP). It is also shown that if X is a metric space and acts on by the action $(x,p){\cdot}q